3.42.40 \(\int ((4-2 e^2) \log (x)+(8-e^2) \log ^2(x)+(6 \log (x)+3 \log ^2(x)) \log (9 x^2)) \, dx\)

Optimal. Leaf size=21 \[ x \log ^2(x) \left (2-e^2+3 \log \left (9 x^2\right )\right ) \]

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Rubi [B]  time = 0.16, antiderivative size = 86, normalized size of antiderivative = 4.10, number of steps used = 16, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2295, 2296, 6688, 12, 2361, 2360, 6742} \begin {gather*} 3 x \log \left (9 x^2\right ) \log ^2(x)+2 \left (8-e^2\right ) x-2 \left (2-e^2\right ) x-12 x+\left (8-e^2\right ) x \log ^2(x)-6 x \log ^2(x)-2 \left (8-e^2\right ) x \log (x)+2 \left (2-e^2\right ) x \log (x)+12 x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 - 2*E^2)*Log[x] + (8 - E^2)*Log[x]^2 + (6*Log[x] + 3*Log[x]^2)*Log[9*x^2],x]

[Out]

-12*x - 2*(2 - E^2)*x + 2*(8 - E^2)*x + 12*x*Log[x] + 2*(2 - E^2)*x*Log[x] - 2*(8 - E^2)*x*Log[x] - 6*x*Log[x]
^2 + (8 - E^2)*x*Log[x]^2 + 3*x*Log[x]^2*Log[9*x^2]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2360

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Log[(c_.)*(x_)^(n_.)]*(e_.) + (d_))^(q_.), x_Symbol] :> Int[E
xpandIntegrand[(a + b*Log[c*x^n])^p*(d + e*Log[c*x^n])^q, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[p
] && IntegerQ[q]

Rule 2361

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.)), x_Symbol] :> With[{u =
IntHide[(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[SimplifyIntegrand[u/x, x], x],
x]] /; FreeQ[{a, b, c, d, e, f, n, p, r}, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (2 \left (2-e^2\right )\right ) \int \log (x) \, dx+\left (8-e^2\right ) \int \log ^2(x) \, dx+\int \left (6 \log (x)+3 \log ^2(x)\right ) \log \left (9 x^2\right ) \, dx\\ &=-2 \left (2-e^2\right ) x+2 \left (2-e^2\right ) x \log (x)+\left (8-e^2\right ) x \log ^2(x)-\left (2 \left (8-e^2\right )\right ) \int \log (x) \, dx+\int 3 \log (x) (2+\log (x)) \log \left (9 x^2\right ) \, dx\\ &=-2 \left (2-e^2\right ) x+2 \left (8-e^2\right ) x+2 \left (2-e^2\right ) x \log (x)-2 \left (8-e^2\right ) x \log (x)+\left (8-e^2\right ) x \log ^2(x)+3 \int \log (x) (2+\log (x)) \log \left (9 x^2\right ) \, dx\\ &=-2 \left (2-e^2\right ) x+2 \left (8-e^2\right ) x+2 \left (2-e^2\right ) x \log (x)-2 \left (8-e^2\right ) x \log (x)+\left (8-e^2\right ) x \log ^2(x)+3 \int \left (2 \log (x) \log \left (9 x^2\right )+\log ^2(x) \log \left (9 x^2\right )\right ) \, dx\\ &=-2 \left (2-e^2\right ) x+2 \left (8-e^2\right ) x+2 \left (2-e^2\right ) x \log (x)-2 \left (8-e^2\right ) x \log (x)+\left (8-e^2\right ) x \log ^2(x)+3 \int \log ^2(x) \log \left (9 x^2\right ) \, dx+6 \int \log (x) \log \left (9 x^2\right ) \, dx\\ &=-2 \left (2-e^2\right ) x+2 \left (8-e^2\right ) x+2 \left (2-e^2\right ) x \log (x)-2 \left (8-e^2\right ) x \log (x)+\left (8-e^2\right ) x \log ^2(x)+3 x \log ^2(x) \log \left (9 x^2\right )-6 \int \left (2-2 \log (x)+\log ^2(x)\right ) \, dx-12 \int (-1+\log (x)) \, dx\\ &=-2 \left (2-e^2\right ) x+2 \left (8-e^2\right ) x+2 \left (2-e^2\right ) x \log (x)-2 \left (8-e^2\right ) x \log (x)+\left (8-e^2\right ) x \log ^2(x)+3 x \log ^2(x) \log \left (9 x^2\right )-6 \int \log ^2(x) \, dx\\ &=-2 \left (2-e^2\right ) x+2 \left (8-e^2\right ) x+2 \left (2-e^2\right ) x \log (x)-2 \left (8-e^2\right ) x \log (x)-6 x \log ^2(x)+\left (8-e^2\right ) x \log ^2(x)+3 x \log ^2(x) \log \left (9 x^2\right )+12 \int \log (x) \, dx\\ &=-12 x-2 \left (2-e^2\right ) x+2 \left (8-e^2\right ) x+12 x \log (x)+2 \left (2-e^2\right ) x \log (x)-2 \left (8-e^2\right ) x \log (x)-6 x \log ^2(x)+\left (8-e^2\right ) x \log ^2(x)+3 x \log ^2(x) \log \left (9 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 21, normalized size = 1.00 \begin {gather*} x \log ^2(x) \left (2-e^2+3 \log \left (9 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - 2*E^2)*Log[x] + (8 - E^2)*Log[x]^2 + (6*Log[x] + 3*Log[x]^2)*Log[9*x^2],x]

[Out]

x*Log[x]^2*(2 - E^2 + 3*Log[9*x^2])

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fricas [A]  time = 0.90, size = 27, normalized size = 1.29 \begin {gather*} 6 \, x \log \relax (x)^{3} - {\left (x e^{2} - 6 \, x \log \relax (3) - 2 \, x\right )} \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(x)^2+6*log(x))*log(9*x^2)+(8-exp(2))*log(x)^2+(-2*exp(2)+4)*log(x),x, algorithm="fricas")

[Out]

6*x*log(x)^3 - (x*e^2 - 6*x*log(3) - 2*x)*log(x)^2

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giac [B]  time = 0.21, size = 62, normalized size = 2.95 \begin {gather*} 6 \, x {\left (\log \relax (3) - 1\right )} \log \relax (x)^{2} + 6 \, x \log \relax (x)^{3} - 2 \, {\left (x \log \relax (x) - x\right )} {\left (e^{2} - 2\right )} - {\left (x \log \relax (x)^{2} - 2 \, x \log \relax (x) + 2 \, x\right )} {\left (e^{2} - 8\right )} + 12 \, x \log \relax (x) - 12 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(x)^2+6*log(x))*log(9*x^2)+(8-exp(2))*log(x)^2+(-2*exp(2)+4)*log(x),x, algorithm="giac")

[Out]

6*x*(log(3) - 1)*log(x)^2 + 6*x*log(x)^3 - 2*(x*log(x) - x)*(e^2 - 2) - (x*log(x)^2 - 2*x*log(x) + 2*x)*(e^2 -
8) + 12*x*log(x) - 12*x

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maple [A]  time = 0.06, size = 27, normalized size = 1.29

method	result	size
norman	\(\left (2-{\mathrm e}^{2}\right ) x \ln \relax (x )^{2}+3 x \ln \relax (x )^{2} \ln \left (9 x^{2}\right )\)	\(27\)
default	\(-x \,{\mathrm e}^{2} \ln \relax (x )^{2}+2 x \ln \relax (x )^{2}+3 \ln \relax (x )^{2} \ln \left (x^{2}\right ) x +6 x \ln \relax (3) \ln \relax (x )^{2}\)	\(38\)
risch	\(6 x \ln \relax (x )^{3}-\frac {3 i \pi \ln \relax (x )^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) x}{2}+3 i \pi \ln \relax (x )^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} x -\frac {3 i \pi \ln \relax (x )^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} x}{2}+6 x \ln \relax (3) \ln \relax (x )^{2}+2 x \ln \relax (x )^{2}-x \,{\mathrm e}^{2} \ln \relax (x )^{2}\)	\(98\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*ln(x)^2+6*ln(x))*ln(9*x^2)+(8-exp(2))*ln(x)^2+(-2*exp(2)+4)*ln(x),x,method=_RETURNVERBOSE)

[Out]

(2-exp(2))*x*ln(x)^2+3*x*ln(x)^2*ln(9*x^2)

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maxima [B]  time = 0.38, size = 76, normalized size = 3.62 \begin {gather*} -{\left (\log \relax (x)^{2} - 2 \, \log \relax (x) + 2\right )} x {\left (e^{2} - 8\right )} - 6 \, x \log \relax (x)^{2} - 2 \, {\left (x \log \relax (x) - x\right )} {\left (e^{2} - 2\right )} + 3 \, {\left ({\left (\log \relax (x)^{2} - 2 \, \log \relax (x) + 2\right )} x + 2 \, x \log \relax (x) - 2 \, x\right )} \log \left (9 \, x^{2}\right ) + 12 \, x \log \relax (x) - 12 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(x)^2+6*log(x))*log(9*x^2)+(8-exp(2))*log(x)^2+(-2*exp(2)+4)*log(x),x, algorithm="maxima")

[Out]

-(log(x)^2 - 2*log(x) + 2)*x*(e^2 - 8) - 6*x*log(x)^2 - 2*(x*log(x) - x)*(e^2 - 2) + 3*((log(x)^2 - 2*log(x) +
2)*x + 2*x*log(x) - 2*x)*log(9*x^2) + 12*x*log(x) - 12*x

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mupad [B]  time = 2.92, size = 20, normalized size = 0.95 \begin {gather*} x\,{\ln \relax (x)}^2\,\left (3\,\ln \left (9\,x^2\right )-{\mathrm {e}}^2+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(9*x^2)*(6*log(x) + 3*log(x)^2) - log(x)*(2*exp(2) - 4) - log(x)^2*(exp(2) - 8),x)

[Out]

x*log(x)^2*(3*log(9*x^2) - exp(2) + 2)

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sympy [A]  time = 0.26, size = 27, normalized size = 1.29 \begin {gather*} 6 x \log {\relax (x )}^{3} + \left (- x e^{2} + 2 x + 6 x \log {\relax (3 )}\right ) \log {\relax (x )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*ln(x)**2+6*ln(x))*ln(9*x**2)+(8-exp(2))*ln(x)**2+(-2*exp(2)+4)*ln(x),x)

[Out]

6*x*log(x)**3 + (-x*exp(2) + 2*x + 6*x*log(3))*log(x)**2

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